Consider n sensors placed randomly and independently with the uniform
distribution in a dβdimensional unit cube (dβ₯2). The sensors have
identical sensing range equal to r, for some r>0. We are interested in
moving the sensors from their initial positions to new positions so as to
ensure that the dβdimensional unit cube is completely covered, i.e., every
point in the dβdimensional cube is within the range of a sensor. If the
i-th sensor is displaced a distance diβ, what is a displacement of minimum
cost? As cost measure for the displacement of the team of sensors we consider
the a-total movement defined as the sum Maβ:=βi=1nβdiaβ, for some
constant a>0. We assume that r and n are chosen so as to allow full
coverage of the dβdimensional unit cube and a>0.
The main contribution of the paper is to show the existence of a tradeoff
between the dβdimensional cube, sensing radius and a-total movement. The
main results can be summarized as follows for the case of the dβdimensional
cube.
If the dβdimensional cube sensing radius is 2n1/d1β and
n=md, for some mβN, then we present an algorithm that uses
O(n1β2daβ) total expected movement (see Algorithm 2 and
Theorem 5).
If the dβdimensional cube sensing radius is greater than
(31/dβ1)(31/dβ1)33/dβ2n1/d1β and n is a natural
number then the total expected movement is
O(n1β2daβ(nlnnβ)2daβ)
(see Algorithm 3 and Theorem 7).
In addition, we simulate Algorithm 2 and discuss the results of our
simulations